Example 5.8 Let \(M=\mathbb {H}\mathrm {P}^{m}\) . In this case the vector bundles \(E\)  and \(F\)  exist globally and \(E\)  is precisely the tautological line bundle. The cohomology ring \(H^{*}(\mathbb {H}\mathrm {P}^{m};\mathbb {Z})\)  is generated by the class \(u=-c_2(E)\) , which also satisfies \((u^{m})\,[\mathbb {H}\mathrm {P}^{m}]=1\) . Moreover, one can show (see [1]}) that the Pontryagin classes of \(T\mathbb {H}\mathrm {P}^{m}\)  are given by
\(p(T\mathbb {H}\mathrm {P}^{m}) = (1+u)^{2m+2}(1+4u)^{-1},\) 
